Abstract
English translation of Moritz Pasch,“Betrachtungen zur Begründung der Mathematik: Zweite Abhandlung,” Mathematische Zeitschrift 25 (1926), pp. 166–171. While insisting on the correctness of his earlier paper on “The Origin of the Concept of Number,” Pasch concedes that he did not provide a foundation for number theory that was “complete in every detail.” He now undertakes to “reconsider and improve” his treatment of two topics: the distinction between restricted and unrestricted common names and the related distinction between (restricted) collections and (unrestricted) sets .
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Notes
- 1.
I say that a mathematical result satisfying the decidability requirement is settled mathematics, while everything else is unsettled . See “The Decidability Requirement” [4]. I had earlier distinguished between “proper” and “improper” calculation: Variable and Function [3], §76. Unsettled mathematics has to rely on settled mathematics, but not vice versa.
- 2.
See Lectures on Modern Geometry [1]. In this work, I start with the notions of point , straight segment , and (bounded) planar surface rather than point, line , and plane. This allows me to define planes and unbounded straight lines. Though I neglect curves entirely in this early book, I take the first steps toward an axiomatic characterization of them in my Prelude to Geometry: the Essential Ideas [6].
- 3.
In a systematic presentation of mathematics, these concepts must precede all others. So they are indispensable to settled mathematics. You thoroughly misunderstand this situation if you treat the natural numbers (or their notation) as given.
- 4.
Perhaps we could identify connections that would allow us to draw together the many topics I develop in the opening sections of Foundations of Analysis [2].
- 5.
[Suppose U is our name for the lines consisting of the things a,b,c. Suppose yesterday I pointed to a, then pointed to b, then pointed to c. These three acts of pointing create a line \(\mathfrak{A}\) with members a,b,c. Today I could create a line distinct from \(\mathfrak{A}\), though with the same members, by pointing to a,b,c in a different order: say, b first, then a, then c. The crucial point, however, is that I could create a line distinct from \(\mathfrak{A}\) by pointing to a,b,c in the same order: a first, then b, then c. This is because today’ s acts of pointing are events distinct from yesterday’ s acts of pointing. Tomorrow I can create yet another line with members a,b,c specified in that order. For Pasch, it is axiomatic that the imaginary person who constructs arithmetic can repeat this process of line formation without limit. (See Section 3.13 of “The Origin of the Concept of Number.”) This means that the name U or the description “line with members a,b,c” will never have an unchangeable denotation. The idealized combinatorial reasoner can always form a new line bearing that name and answering to that description.]
- 6.
[Pasch probably means no. 5, not no. 4. If a name is collective, then Pasch’ s combinatorial reasoner is able to specify each thing to which the name applies. These specifications create a thing, a line , to which the bearers of the name belong. The reasoner then uses implicit definition to introduce the extension of that line, the collection of the things specified. The reasoner cannot specify each thing to which an unrestricted common name applies and, so, cannot form a line of those things. Instead, he must proceed directly to an implicit definition .]
- 7.
* [Reference [6] (monograph) included the papers translated above as “Rigid bodies in geometry” and “Prelude to geometry: The essential ideas.”]
References
* [Reference [6] (monograph) included the papers translated above as “Rigid bodies in geometry” and “Prelude to geometry: The essential ideas.”]
Pasch, Moritz. 1882. Lectures on modern geometry. Leipzig: B.G. Teubner.
Pasch, Moritz. 1909. Foundations of analysis. Leipzig: B.G. Teubner.
Pasch, Moritz. 1914. Variable and function. Leipzig: B.G. Teubner.
Pasch, Moritz. 1918. The decidability requirement. Jahresbericht der deutschen Mathematiker-Vereinigung 27:228–232.
Pasch, Moritz. 1919/1921. The origin of the concept of number. Archiv der Mathematik und Physik 28:17–33 and Mathematische Zeitschrift 11:124–156.
Pasch, Moritz. 1922. Prelude to geometry: the essential ideas. Leipzig: Felix Meiner.
Pasch, Moritz. 1924. Reflections on the proper grounding of mathematics. Mathematische Zeitschrift 20:231–240.
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Pollard, S. (2010). Reflections on the Proper Grounding of Mathematics II. In: Pollard, S. (eds) Essays on the Foundations of Mathematics by Moritz Pasch. The Western Ontario Series in Philosophy of Science, vol 83. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9416-2_14
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